Predicting the combined effects of case isolation, safe funeral practices, and contact tracing during Ebola virus disease outbreaks

Background The recent outbreaks of Ebola virus disease (EVD) in Uganda and the Marburg virus disease (MVD) in Ghana reflect a persisting threat of Filoviridae to the global health community. Characteristic of Filoviridae are not just their high case fatality rates, but also that corpses are highly contagious and prone to cause infections in the absence of appropriate precautions. Vaccines against the most virulent Ebolavirus species, the Zaire ebolavirus (ZEBOV) are approved. However, there exists no approved vaccine or treatment against the Sudan ebolavirus (SUDV) which causes the current outbreak of EVD. Hence, the control of the outbreak relies on case isolation, safe funeral practices, and contact tracing. So far, the effectiveness of these control measures was studied only separately by epidemiological models, while the impact of their interaction is unclear. Methods and findings To sustain decision making in public health-emergency management, we introduce a predictive model to study the interaction of case isolation, safe funeral practices, and contact tracing. The model is a complex extension of an SEIR-type model, and serves as an epidemic preparedness tool. The model considers different phases of the EVD infections, the possibility of infections being treated in isolation (if appropriately diagnosed), in hospital (if not properly diagnosed), or at home (if the infected do not present to hospital for whatever reason). It is assumed that the corpses of those who died in isolation are buried with proper safety measures, while those who die outside isolation might be buried unsafely, such that transmission can occur during the funeral. Furthermore, the contacts of individuals in isolation will be traced. Based on parameter estimates from the scientific literature, the model suggests that proper diagnosis and hence isolation of cases has the highest impact in reducing the size of the outbreak. However, the combination of case isolation and safe funeral practices alone are insufficient to fully contain the epidemic under plausible parameters. This changes if these measures are combined with contact tracing. In addition, shortening the time to successfully trace back contacts contribute substantially to contain the outbreak. Conclusions In the absence of an approved vaccine and treatment, EVD management by proper and fast diagnostics in combination with epidemic awareness are fundamental. Awareness will particularly facilitate contact tracing and safe funeral practices. Moreover, proper and fast diagnostics are a major determinant of case isolation. The model introduced here is not just applicable to EVD, but also to other viral hemorrhagic fevers such as the MVD or the Lassa fever.


Basic model compartments
A population of N individuals is assumed. Without interventions, the compartmental model (Fig 1) follows the time change in the number of individuals being susceptible (S), latently infected (E Sum ) consisting of n E sub-states (E k , k = 1, . . . , n E ), prodromal (P Sum ) consisting of n P sub-states (P k , k = 1, . . . , n P ), fully infectious at home (I Sum ) consisting of n IIso sub-states (I k,Iso , k = 1, . . . , n IIso ), finally being recovered (R), or dead, where individuals first await funeral (F ) before being buried (B F ). To model intervention, additional compartments are added as described below.

Durations
The average duration of the latent, and prodromal phases are denoted by D E and D P , respectively. The duration of the fully infectious disease can in principle be different for hospitalized and isolated cases due to administered treatments. The durations of the fully infectious period for infections at home, in hospital, and in isolation are, respectively, D IHome , D IHosp , and D IIso .
Infected individuals have to first progress through a number of equivalent latent sub-states. Hence, the average duration in each latent sub-state is D E /n E , and individuals leave the latent sub-states at rate As a consequence of modeling sub-states the duration in the total latent phase is no longer exponentially distributed, but Erlang distributed with mean D E and variance D E /n E . The advantage is that D E and n E can be chosen such that they fit empirical estimates in terms of mean and variance. After the latent phase, infected individuals progress through prodromal, and fully infectious sub-states in a similar fashion and the rates are, respectively γ := n P D P , δ Home := n IHome D IHome , δ Hosp := n IHosp D IHosp , and δ Iso := n IIso D Iso . (1b)

Course of the disease
After the prodromal stage all infections become fully infectious. In addition, a fraction f Iso gets diagnosed with EVD and isolated. This fraction is assumed to be time dependent, i.e., f Iso (t), reflects that infected will not immediately get diagnosed and isolated at the beginning of the EVD outbreak. Another fraction f Hosp (t) becomes hospitalized but remains undiagnosed with EVD. The remaining fraction f Home (t) = 1 − f Iso (t) − f Hosp (t) is neither diagnosed nor hospitalized, i.e., the infected individuals stay at home (without proper medical care). Individuals do not die from EVD during the latent or prodromal phase. However, fractions f Dead , and f (Home) Dead of symptomatic infections, respectively isolated, hospitalized, and at home, are lethal. Infections not treated in isolation are more likely to be lethal (particularly f

Transmission during funerals
Unlike with other diseases, deceased individuals are highly contagious. In fact, EVD can spread during funeral ceremonies (F ), which is a recognized concern [1]. If isolated individuals die, they will be buried safely (B Iso ), i.e., transmission does not occur after death. The EVD infection of individuals that die at home or in hospital might be properly recognized after death, in which case they also receive a safe funeral. It is assumed that fractions d Home and d Hosp of individuals that died at home or in hospital receive a safe funeral.

Contact tracing
To model contact tracing, we have to distinguish between infections which are never traced back and not isolated and those that will get isolated and traced back.
Contact tracing can help to diagnose more individuals by investigating contacts of the diagnosed cases. In a compartmental model, contact tracing can only be modeled approximately, because infections are not modeled on an individual basis. Contact tracing is not instantaneous. It takes an average duration until the tracing back becomes effective. To adequately capture this in a compartmental model, we have to introduce new compartments (denoted E * , P * , I ( * ,Home) , I ( * ,Hosp) ) for infections which will become traced back after an average duration 1/α. Furthermore, compartments for infections which are isolated after being traced back (Ẽ, P ) need to be added to the model (cf. Fig 1). In the fully infectious phase, we no longer distinguish between isolated cases which where found by contact tracing and those diagnosed for other reasons (I Iso ). The model accounts for sub-states of all phases of infected individuals, i.e., those who (i) are never traced back, (ii) are not yet traced back (transient phase), and (iii) are traced back. The respective compartments of those who will get traced back (transient phase) are denoted by E * k (k = 1, . . . , n E ), P * k (k = 1, . . . , n P ), I ( * ,Home) k (k = 1, . . . , n IHome ), and I ( * ,Hosp) k (k = 1, . . . , n IHosp ). The compartment of those which are successfully traced back are denoted byẼ k (k = 1, . . . , n E ), andP k (k = 1, . . . , n P ). The transition rates in the respective compartment are as for infections that are not traced back (given by Eq. 1a-1b)

Onset of interventions
Importantly, intervention measures (case isolation, safe funeral, and contact tracing) will be established only some time after the occurrence of the initial cases. In other words, there is an initial period of t Iso days, after which all measure are implemented.
Regarding isolation and hospitalization we assume and

Cumulative numbers of infections
The total number of latently infected which will not get traced back is while that of latent infections, which will be traced back successfully is and that of the latent infection, which were traced back successfully in the latent stage is Likewise the number of prodromal infections that will not get traced back successfully, will be traced back successfully, and were traced back are, respectively, andP Finally, the number of fully infectious individuals which are in isolation (because they were diagnosed with EVD or successfully traced back) is given by

Case Isolation
Case isolation is intertwined with contact tracing. Individuals that are successfully traced back will be isolated during any phase of the disease. All individuals, which are successfully traced back, will be in isolation during the fully infectious phase before they eventually recover or die. Additionally, as described above, a fraction f Iso (t) (which is time dependent) of fully infectious individuals, not subject to successful back tracing, are EVD-diagnosed and isolated. Isolation lasts until recovery or death. Isolated patients are treated in quarantine wards and do no longer contribute to transmitting the virus. However, quarantine wards have a maximum capacity Q max . The number of isolated cases in excess of the capacities, can no longer be perfectly isolated, in fact only a fraction p Excess of infectious contacts is prevented compared to hospital conditions. The number of infections subject to isolation is given by The proportion of individuals subject to isolation, which are perfectly isolated is The remaining proportion 1 − q(t) of these individuals are isolated imperfectly, i.e., compared to hospitalized patients their contacts are reduced by a fraction (1 − p Excess ). Importantly, if they die, they will be sent to safe funeral.
Eventually not every individual that should be traced back, will be traced back. The number of individuals that are supposed to be traced back is Thus, the probability that an individual that should be traced back, is actually traced back is given by if t ≥ t Iso and C(t) > C max , December 28, 2022 4/17 where, C max is the maximum number of individuals that can be traced back.

Contact rates
Susceptibles encounter infected individuals (not in isolation) randomly. The relative contagiousness of prodromal individuals, fully infectious individuals at home, fully infectious individuals in hospital, and deceased individuals at unsafe funerals (who are the most contagious) are c P , c IHome , c IHosp , and c F , respectively (typically c P ≤ c IHosp ≤ c IHome ≤ c F ). Individuals that will never get traced back and those who will get traced back are equally contagious in the respective disease phases before they get isolated. The contact rates depend on the basic reproduction number R 0 described in detail below. This number is determined in a susceptible population before any disease-control interventions are implemented. Note that the fractions of infections which are hospitalized or isolated change over time. Writing f Hosp = f Hosp (0) and f Home = f Home (0) the contact rates in the respective infectious phases are and where

Force of infection
During the prodromal and the fully infectious phase, individuals can transmit the virus, the same is true for corpses before being buried (F ). Without intervention, transmission is related to the contact rates of the infected in the respective phases. These are modified by case isolation and safe funeral procedures. Isolated individuals (P Sum , I Sum ) cannot transmit the disease until the capacity of the quarantine wards are exhausted. Individuals that cannot be isolated in quarantine wards (a fraction 1 − q(t) ofP Sum and I (Iso) Sum ) are imperfectly isolated (the infectious contacts reduce by a fraction p Excess ).
All other individuals in the infectious phases (P Sum , P * Sum , I Therefore, the total force of infection is December 28, 2022 5/17 Newly infected individuals will either be subject to contact tracing, i.e., they will be successfully traced back during their infection, or they will not be successfully traced or traced at all. Depending on the outcome of contact tracing, newly infected individuals are either subsumed by the compartments E * 1 or E 1 . Consequently the total force of infection is split into two parts corresponding to the force of infection experienced by susceptible that will be successfully traced back λ * and by those that will not. Hence, Fully infectious individuals not subject to back tracing (I (Home) Sum , I Sum , F ) only cause infections which will not be traced back. These infections are attributed to λ.
If an individual is successfully traced back, i.e., it was found based on tracing the original infectious contact, this individual might have caused secondary infections, which will also be subject to back tracing. Importantly, only a fraction f Tr of secondary infection can be successfully traced back. Furthermore, the capacity of back tracing is limited, such that only a fraction c(t) of secondary infections, which could be traced back actually can be traced back. Altogether only a fraction f Tr c(t) of secondary infections of individuals subject to successful back tracing (P * Sum , I ( * ,Home) Sum , I ( * ,Hosp) Sum ) will also be traced back and attributed to λ * . Infections that cannot be properly isolated because the capacity of quarantine wards are reached ((1 − q(t))P Sum , (1 − q(t))I (Iso) Sum ) deserve particular attention. Again only a fraction f Tr c(t) of secondary infections caused by them will be subject to back tracing. However, their contacts are limited by an amount 1 − p Excess .
Prodromal cases also deserve particular attention. While secondary cases of prodromal infections subject to back tracing (P * Sum ,P Sum ) are traced back with probability (f Tr c(t)), back tracing of cases subsumed by P Sum are not so obvious. Namely, a fraction f Iso (t) of prodromals not subject to back tracing yet, will get diagnosed and isolated later. Consequently, they will be subject to back tracing later. Therefore, a fraction of (f Iso (t)f Tr c(t)) of the contacts of infections subsumed by P Sum will be successfully traced back and attributed to λ * .
Consequently, the force of infection experienced by suceptibles that will be subject to successful back tracing is while that experienced by those that will not be subject to successful back tracing is

Differential Equations
From the above, the system of differential equations describing the dynamics of the spread of EVD follows. The change in the number of susceptibles is given by The numbers of infected individuals in the latent sub-stages not subject to successful back tracing change according to and while the dynamics of those of latently infected individuals subject to successful back tracing are and The numbers of latently infected individuals, who are traced back and isolated change according to and The numbers of prodromal individuals change according to and The numbers fully infectious individuals at home or in hospital, who will not be traced back, change according to dI 1,Home (t) dt = γf Home (t)P n P (t) − δ Home I 1,Home (t), (22a) dI k,Home (t) dt = δ Home I k−1,Home (t) − δ Home I k,Home (t) for 2 ≤ k ≤ n IHome , (22b) The numbers fully infectious individuals in isolation change according to The fully infectious individuals at home and in hospital who will get traced back change according to dI * k,Home (t) dt = δ Home I * k−1,Home (t) − (α + δ Home )I * k,Home (t) for 2 ≤ k < n IHome , (25b) dI * k,Home (t) dt = δ Home I * k−1,Home (t) − αI * k,Home (t) for k = n IHome . (25c) Finally, the number of recovered individuals changes according to The number of deceased cases having an unsafe funeral (during which the corpse are highly contagious) changes according to Dead I n I Home ,Home (t) Dead I n I Hosp ,Hosp (t) − ϕF (t). (28) December 28, 2022 8/17 The number of corpse, which are buried after a funeral changes according to

Basic reproduction number
For convenience the model was parameterized in a way that the basic reproduction number R 0 enters the model as a parameter. Recall, that R 0 is the number of the secondary infections caused by an average single infected individual in a virgin population without any control interventions. However, according to [2], R 0 is also the spectral radius (maximum absolute eigenvalues) of the next-generation matrix (NGM). In other words, if G denotes the NGM, Therefore, in the parameter R 0 has to emerge also from the NGM, which will yield the correct definition of the contact rates.
To derive the NGM, first the reduced system of ODEs, needs to be considered and linearized. The reduced system retains only the equations from the original system, which describe the states of infected individuals that are relevant in the absence of interventions. Let X X X := [E E E, P P P , I I I Home , I I I Hosp , F ] be the vector of all variables in the reduced system, where E E E, P P P , I I I Home , I I I Hosp comprise several states each E E E = (E 1 , . . . , E n E ), P P P = (P 1 , . . . , P n P ), I I I Home = (I 1,Home , . . . , I n I Home ,Home ), I I I Hosp = (I 1,Hosp , . . . , I n I Hosp ,Hosp ). (In the following, we use brackets to denote vectors or matrices, whose entries are vectors or matrices.) The reduced system first splits into a vector F describing disease transmission and a vector V describing the progression of the disease. The Jacobian of these two vectors needs to be derived at the disease free state X X X 0 . The NGM is defined as The vector describing transmission is where and where (34a)-(34d) are vectors, containing the partial derivative with regard to the various variables (compartments) and 0 0 0 = (0, . . . , 0), β β β P = (β P , . . . , β P ), β β β IHome = (β IHome , . . . , β IHome ), and β β β IHosp = (β IHosp , . . . , β IHosp ), are vectors of dimensions n E , n P , n IHome , and n IHosp , respectively. Hence, the Jacobian of F, denoted by F , is where 0 0 0 denote vectors with all elements being 0 of appropriate dimension. The vector F describes transmission, i.e., new infections. The vector V describes the disease progression between all infected states. For the latent state E 1 according to (20), the component describing the disease progression is For all other states the entries of V follow directly from the relevant components of the ODE system, (20a)-(20b), (21a)-(21b), (22a)-(22b), (23a)-(23b), and (28) by ignoring all terms, which correspond to interventions. In particular, one obtains where γf Home P n P (t) − δ Home I 1,Home (t) . . .
V F = δ Home bI n I Home ,Home (t) + δ Hosp aI n I Hosp , The Jacobian of V, denoted by V (X X X 0 ) consists of the partial derivatives of V with respect to all states evaluated at X X X 0 . For further computation it is convenient to split the Jacobian into blocks corresponding the states E E E, P P P , I I I Home , I I I Hosp , and F .
The inverse of V (X X X 0 ) needs to be calculated. This is conveniently done, by successively applying a block-wise inversion formula. From (37) one recognizes a blockwise structure, namely define where we define and The remaining blocks are defined as B 1 := ∂V P P P (X X X) ∂I I I Home ∂V P P P (X X X) ∂I I I Hosp A 2 := ∂V I I I Home (X X X) A 3 := ∂V I I I Hosp (X X X) and To invert V we successively apply a block-wise inversion matrix four times, to the December 28, 2022 13/17 following matrices and First, using block-wise inversion for D 2 yields Because B 3 = 0 0 0, (41) simplifies to The inverse matrix of A 3 and D 3 that occur in the above equation are straightforwardly calculated to be and December 28, 2022 14/17 from which D −1 2 follows explicitly. Namely, Next, block-wise inversion of D 1 yields where D −1 2 is given by (44) and A −1 2 is derived similarly as A −1 3 , namely The product −D −1 2 C 2 A −1 2 is straightforwardly derived. Similarly, D −1 0 is derived as where A −1 1 is derived in the same way as A −1 2 or A −1 3 , namely The product −D −1 1 C 1 A −1 1 follows immediately. Finally, the matrix V is inverted as where A −1 0 is derived to be Although −D −1 0 C 0 A −1 0 follows straightforwardly, it is not necessary to calculate it explicitly. The reason is that we need only to calculate the eigenvalues of the product From (35) and (37) it follows that only the first row of G contains non-zero elements. Thus, the only eigenvalue different from zero is the first entry of the matrix G, i.e., the product of the first row of F and the first column of V . A tedious but simple calculation shows R 0 := ρ(G) = n P β P γ + n IHome β IHome f Home δ Home + n IHosp β IHosp f Hosp δ Hosp + β F u ϕ . (52a) Using the definition of γ, δ Home , δ Hosp , and ϕ (1b) yields

Computation of contact rates
The contact rates are proportional to the product of R 0 and the relative contagiousness in the respective phase of the disease (c P , c I , c IHosp , and c F ), i.e., Deriving the appropriate proportional constant K yields 15a, 15b, 15c, and 15d. Remember, the basic reproduction number R 0 has to emerge as the spectral radius of the NGM. Combining (53) and (52b), gives which implies Substituting (55) into (53) yields (53a), (53b), (53c) and (53d).